THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

SANTA  BARBARA 

COLLEGE 

PRESENTED  BY 

William  E.  Roberts 


THE 

TEETH  OF  SPUR  WHEELS; 

THEIR  CORRECT  FORMATION 

IN 

THEORY    A^D    PRACTICE. 

BY  PROF.  C.  W.  MAcCORD. 


HARTFORD,   CONN.  : 

PUBLISHED   BY   THE  PRATT  &  WHITNEY  COMPANY, 
RS  ov   MACHINISTS'   TOOLS,   GUN   AND  SBWING   MACHINE   MACHINERY,  &c.,  &C. 


PREFACE. 


As  a  preliminary  to  the  description  of  the  machines  for  the  accurate 
formation  of  cutters  for  spur  wheels,  given  in  the  second  part  of  this 
treatise,  it  seemed  appropriate  to  explain  the  manner  of  laying  out  the 
teeth.  And  in  the  belief  that  it  may  be  acceptable  to  many  who  are 
interested  in  the  subject,  we  have  endeavored  to  give,  in  as  simple,  clear 
and  brief  a  manner  as  possible,  the  reasons  and  the  proof  of  every  step  in 
the  construction.  The  method  of  drawing  rolled  curves  by  means  of 
tangent  arcs,  which  we  believe  to  be  the  most  accurate  and  expeditious 
known,  may  be  new  to  some;  and  this,  as  well  as  Prof.  Rankine's  elegant 
graphic  processes  relating  to  circular  arcs,  will  be  found  applicable  to 
many  other  purposes,  and  exceedingly  useful.  To  which  we  may  add, 
finally,  the  hint  that  the  same  is  true  of  the  principles  and  the  methods 
made  use  of  in  the  demonstrations. 

C.  W.   MAC-CORD. 
STEVENS  INSTITUTE  OF  TECHNOLOGY, 
Hoboken,  N.  J.,  Jan.  28,  1881. 


\  VS  (^5  UNIVERSITY  OF  CALIFORNIA 

SANTA  BARBARA  L^LLESE  LIBRARY 

73256 

PART     I. 


THE    TEETH    OF   WHEELS. 


GENERAL  PRINCIPLES. 

The  proper  action  of  many  pieces  of  mechanism  depends  so  largely 
upon  that  of  spur  wheels,  that  any  means  of  effecting  a  radical  improve- 
ment in  the  making  of  such  wheels  cannot  but  be  of  interest  and  im- 
portance. 

There  was  a  time  when  the  teeth  of  wheels  were  made  in  rude  hap- 
hazard ways,  of  almost  any  shapes  that  would  permit  them  to  engage, 
with  a  mistaken  idea  that  they  would  wear  themselves  into  correct  forms. 
The  machine  was  expected  not  only  to  do  its  own  proper  work,  but  partly 
to  finish  itself;  small  wonder,  then,  that  it  failed  to  do  either  of  these 
things  well.  Naturally,  these  crude  methods  gave  place  to  better  ones.  The 
mechanician  perceived  the  necessity  of  greater  care  in  making  the  teeth 
of  proper  form ;  the  mathematician  soon  became  interested  in  the  problem 
of  determining  what  forms  were  proper,  and  the  results  of  their  combined 
efforts,  leave  little  to  be  desired  in  relation  to  the  latter. 

And  as  little  would  seem  to  be  left  in  regard  to  the  former  after  the 
introduction  of  the  gear-cutting  engine,  by  which,  if  the  milling  cutter  be 
of  the  correct  outline,  all  the  teeth  of  a  wheel  are  made  with  the  utmost 
regularity  and  precision.  But  on  closer  consideration,  it  will  be  seen  that 
something  is  yet  lacking  in  reference  to  the  formation  of  the  cutter  itself. 


It  is  one  thing  to  know  what  its  outline  should  be,  but  quite  another  thing 
to  make  it  so. 

The  process  most  extensively  employed  involves,  1st,  the  laying  out  of 
the  required  curve  ;  2nd,  the  filing  of  a  template  to  that  exact  form,  and, 
3rd,  the  turning  of  the  cutter  to  fit  the  template. 

In  some  cases  a  specially-contrived  apparatus  has  been  used  for  me- 
chanically tracing  the  curve  by  continuous  motion,  but,  until  recently,  the 
two  remaining  steps  have  been  executed  by  hand,  which  makes  the  per- 
fectly accurate  formation  of  a  cutter,  especially  if  it  be  a  small  one,  vexy 
difficult,  and  its  exact  duplication  still  more  so. 

The  time  has  now  come  when  all  this  ought  to  be  changed.  No  one 
who  considers  for  a  moment  the  vast  numbers  of  accurate  machines  em- 
ployed in  the  various  industrial  arts,  and  of  others  equally  accurate,  em- 
ployed in  making  them,  can  fail  to  perceive  the  advantages  over  the 
system  above  described,  of  one  in  which  the  template  is  not  merely  lined 
out,  but  cut  out  to  the  true  form,  and  the  contour  of  the  milling  cutter,  be 
it  large  or  small,  is  made  to  correspond  to  that  of  the  template,  by  mechan- 
ism nearly  automatic.  Of  such  a  system,  and  of  the  means  by  which 
these  results  are  effected,  we  propose  to  give  a  detailed  description.  Before 
entering  upon  this,  however,  we  shall  briefly  explain  the  principles  upon 
which  the  correct  forms  of  the  teeth  depend,  and  the  method  of  laying 
out  epicycloidal  teeth  in  outside  gear. 

GRAPHIC    REPRESENTATION     OF     MOTION. 

The  motion  of  a  point  at  any  instant  may  be  represented  in  magnitude 
and  direction  by  a  right  line. 

It  is  true  that  the  path  of  the  point  may  be  a  curve  of  any  kind ;  but 
at  any  given  instant  it  can  occupy  but  one  position,  and  its  direction  "Will 


be  that  of  the  tangent  to  the  path  at  that  point.  The  length  of  that  tan- 
gent may,  evidently,  be  made  to  indicate  the  velocity;  therefore  the  motion 
is  fully  represented. 

COMPOSITION    AND    RESOLUTION    OF    MOTION. 

The  composition  (or  finding  the  resultant)  of  two  motions,  is  effected 
as  shown  in  Fig.  1.     Suppose  the  point  A,  to  receive  simultaneously  two 


Fig.  1. 

impulses,  the  motions  due  to  which  are  represented  in  velocity  and  direc- 
tion by  AB,  AC.  Draw  BD  parallel  to  AC,  and  CD  parallel  to  AB :  then 
AD,  the  diagonal  of  the  parallelogram  thus  formed,  will  represent  the  re- 
sultant motion  in  both  direction  and  velocity.  That  is  to  say,  the  point  A 
will  go  to  D,  in  the  same  time  in  which  it  would  have  reached  either  B  or 
C,  had  it  received  but  one  of  the  impulses. 

Evidently,  if  one  component  and  the  resultant  be  known,  the  other 
component  may  be  found  in  a  similar  manner.  If,  for  instance,  we  know 
that  AD  is  the  resultant  of  two  components,  one  of  which  is  AC,  draw 
CD;  then  the  other  component  must  have  the  direction  AB  parallel  to  CD, 
and  its  magnitude  is  found  by  drawing  DB  parallel  to  AC. 

The  resolution  of  motion  is  the  converse  of  composition  :  thus,  it  is  evi- 
dent that  the  motion  AD  in  Fig.  1  may  be  separated  into  the  components 


from  which  it  was  derived,  by  drawing  the  parallels  DC,  AB,  in  one  direc- 
tion, and  DB,  AC,  in  the  other;  by  which  the  original  parallelogram  is 
reconstructed. 

But  again,  AD  may  be  the  diagonal  of  a  great  number  of  other 
parallelograms :  from  which  we  see  that  a  given  motion  may  be  resolved 
into  two  components,  having  any  directions  we  please  to  assign. 

ANGULAR    VELOCITY    AND    VELOCITY    RATIO. 

This  term  angular  velocity  is  applied  only  to  circular  motion,  like  that 
of  a  wheel  revolving  on  its  axis.  Every  point  in  the  revolving  body  turns 
through  the  same  angle  in  the  same  time,  whatever  be  its  distance  from 
the  axis:  thus  in  Fig.  2,  the  point  A  goes  to  D,  in  the  same  time  that  it 


Fig.  2. 

takes  the  point  B  to  reach  E.  Clearly  the  arcs  AD,  BE,  represent  the 
linear  velocities  of  the  moving  points;  but  AD  is  as  many  times  greater 
than  BE,  as  AC  is  greater  than  BC.  If  then  we  divide  AD  by 


BE  by  BC,  either  quotient  may  be  taken  as  the  measure  of  the  angular 
motion  represented  by  A  CD;  or,  in  general,  we  say  that 

linear  velocity 

angular  velocity  = —       -. 

radius 

If  we  consider  the  motion  of  a  revolving  point  at  a  single  instant 
only,  its  direction  is  that  of  the  tangent  to  its  circular  path.  In  represent- 
ing it,  then,  we  set  off  the  linear  velocity  perpendicular  to  the  radius 
through  the  point,  as  AM,  BN.  Drawing  CM,  it  will  be  observed  that  the 
angle  ACJIfis  not  the  same  as  A  CD.  Nor  should  it  be,  since  the  former 
represents  what  is  happening  at  a  given  instant,  the  latter  a  motion  con- 
tinuing through  a  period  of  definite  duration. 

The  velocity  ratio  of  two  revolving  bodies,  at  any  instant,  is  simply  the 
quotient  obtained  by  dividing  the  angular  velocity  of  one  by  that  of  the 
other,  at  the  given  instant.  If  this  quotient  be  the  same  at  every  instant, 
the  velocity  ratio  is  said  to  be  constant;  as  in  the  case  of  two  pulleys 
driven  by  a  belt  which  does  not  slip — if  one  be  half  the  size  of  the  other, 
it  will  always  turn  twice  as  fast,  whether  the  actual  speed  be  uniform 
or  not. 


DETERMINATION     OF     VELOCITY     RATIO. 

In  Fig.  3,  C  and  D  are  fixed  centers,  about  which  turn  the  two  curved 
levers,  CH,  DK,  which  touch  each  other  at  A.  Through  this  point  draw 
TV1  the  common  tangent,  and  -<?WVthe  common  normal,  of  the  two  curves, 
and  also  AC  and  AD,  the  radii  of  contact.  If  CH  turn  in  the  direction 
indicated  by  the  arrow,  DK  will  be  driven  in  the  opposite  direction ;  we 
now  wish  to  find  the  angular  velocity  ratio  of  the  two  motions. 


8 

The  point  A,  of  the  driver  CH,  must  move  in  a  direction  perpendicular 
to  CAy  and  we  will  suppose  its  linear  velocity  at  the  instant  to  be  repre- 
sented by  AB,  which  may  be  resolved  into  the  components  AP,  AM;  the 
first  is  the  effective  component  of  rotation,  the  latter  being  the  sliding 
component. 


Fig.  3. 

The  point  A,  of  the  follower  D K,  must  go  in  a  direction  perpendicular 
to  AD,  and  its  velocity  AL  must  be  such  as  to  have  the  same  normal  com- 
ponent AP,  the  tangential  or  sliding  component  being  AR. 

We  perceive,  then,  that  when  one  revolving  piece  drives  another  by 
contact,  the  motion  is  transmitted  in  the  direction  of  the  common  normal, 
which  is  therefore  called  the  line  of  action. 

Now  draw  CE  and  DF  perpendicular  to  NN,  CD  the  line  of  centrrs, 


cutting  NN  at  7,  also  CB,  DL.     We  shall  thus  have  three  pairs  of  similar 
triangles,  viz. : 

CAE  is  similar  to  ABP, 
DAF  "  "  ALP, 
CIE  "  "  DIP. 

Let  the  angular  velocities  of  the  driver  and  the  follower  respectively 

lin.  vel. 
be  represented  by  v,d ' ;  then  since         ang.   vel.—          — , 

radius 
we  shall  have 

AB  AP 

CA'       CE'' 

and 

AL  AP 

qj    __j,  


DA  DF 

whence 

v         DF  DI 


'    v'         CE  Cf 

That  is  to  say:  The  angular  velocities  are  inversely  proportional  to  the 
perpendiculars  let  fall  from  the  centers  of  motion  upon  the  line  of  action. 

Or  otherwise :  The  angular  velocities  are  inversely  proportional  to  the 
segments  into  which  the  line  of  centers  is  cut  by  the  line  of  action. 


CONDITION     OF     A     CONSTANT     VELOCITY     RATIO. 

If  it  be  required  that  the  velocity  shall  remain  constant,  the  second  of 
the  values,  above   deduced,  indicates  a  condition  which  the  curves  must 


10 

satisfy,  viz. :  they  must  be  of  such  forms  that  their  common  normal  shall 
always  cut  the  line  of  centers  at  the  same  point.  For  the  line  of  centers  is  of 
fixed  length,  whence  the  segments  into  which  it  is  cut  must  always  be  the 
same,  in  order  to  maintain  a  constant  ratio. 

Now  the  outlines  of  the  teeth  of  spur  wheels  must  be  curves  which 
O//7/ transmit  rotation  with  a  constant  velocity  ratio,  and  it  will  readily  be 
seen  that  parts  of  the  curved  levers  in  the  immediate  neighborhood  of  the 
point  of  contact  A,  Fig.  3,  might  be  of  the  forms  proper  for  teeth  of  wfctrels 
whose  centers  are  Cand  D.  In  the  figure,  the  sliding  components  of  ^he 
motions  AS,  AL,  are  respectively  AM,  AR;  these  lie  in  the  same  direc- 
tion, but  are  not  equal ;  so  that  at  this  instant,  the  levers  are  sliding  upon 
each  other  with  the  velocity  RM,  equal  to  their  difference.  There  would 
be  no  sliding  if  these  were  of  the  same  length  as  well  as  in  the  same  direc- 
tion ;  but  since  the  normal  component  AP,  is  the  same  for  each  motion, 
this  can  only  happen  when  the  resultants  AJB,  AL,  also  coincide.  And 
these  being  perpendicular  to  the  radii  of  contact,  cannot  coincide  unless 
CA,  AD,  lie  in  one  right  line,  which  must,  of  course,  be  CD.  That  is  to 
say,  there  will  be  more  or  less  sliding,  except  at  the  instant  when  the  point  of 
contact  is  on  the  line  of  centers. 


NATURE    OF    ROLLING    CONTACT. 

The  finding  of  curves  which  will  satisfy  the  above  condition,  and  also 
the  best  means  of  drawing  them,  depend  upon  a  feature  of  perfect  rolling 
contact,  best  seen  by  a  study  of  that  which  is  not  perfect.  The  polygon, 
in  Fig.  4,  rolls  along  the  fixed  right  line  with  a  hobbling  motion ;  the  point 
A  is  at  rest,  and  the  whole  figure  turns  about  it  as  a  center  until  B  comes 
into  LM  at  D,  then  about  13,  and  so  on  ;  the  perimeter  of  the  polygon 
measuring  itself  off  upon  the  line.  If  the  number  of  sides  be  increased, 


11 


LAD  M 

Fig.  4 

the  hobbling  will  be  diminished,  and  if  the  number  become  inconceivable, 
it  will  become  imperceptible.  The  broken  outline  then  becomes  the 
dotted  curve,  tangent  to  the  line,  and  the  change  from  one  center  of  rota- 
tion to  another  goes  on  continuously.  But  this  does  not  alter  the  facts, 
that  at  any  instant  the  point  of  contact  is  at  rest,  and  that  every  point  in 
the  figure  is  at  the  instant  turning  about  that  point  of  contact  as  a  fixed 
center. 

DRAWING    OF    ROLLED    CURVES. 

In  pig.  5,  AA  is  a  curved  ruler  fixed  to  the  drawing  board,  and  BB  is 
a  free  one  rolling  along  it.  Let  a  pencil  be  fixed  to  and  carried  by  the 
latter,  either  in  the  contact  edge,  as  at  Z>,  or  at  any  distance  from  it,  as  at 
E.  At  the  instant  the  rulers  are  in  contact  at  P,  the  motion  of  D 
is  in  the  direction  DF,  perpendicular  to  DP,  the  contact,  radius.  DF, 
then,  is  tangent  to  the  path  of  D,  traced  as  the  ruler  BB  rolls:  but  it  is  also 
tangent  to  the  circular  arc  whose  center  is  D  and  radius  PD,  consequently 
the  path  of  D  is  also  tangent  to  that  arc.  Let  the  arcs  PC,  Po  of  BB, 


12 


be  equal  to  the  arcs  Pd ',  Po'  of  AA,  then  cD  will  be  contact  radius  when  c 
reaches  d ' ,  and  oD  when  o  reaches  o' .  If,  then,  we  describe,  with  these 
radii,  circular  arcs  about  c1  and  o' ,  the  curve  traced  by  D  will  be  tangent  to 
those  arcs ;  and  that  traced  by  E  will  be  tangent  to  arcs  about  the  same 
centers  with  cE  and  oE  as  radii. 

Curves  thus  described,  by  points  carried  by  one  line  which  rolls  upon 
another,  are  called  rolled  curves  or  epitrochoids ;  and  the  drawing  of  a  series 
of  tangent  arcs  as  above  explained  is  the  readiest  and  most  reliable  method 
of  laying  them  out.  The  line  which  carries  the  tracing  point  is  called  the 
describing  line,  and  the  one  in  contact  with  which  it  rolls  is  called  the  base 
liffe;  either  of  these  may  be  straight,  or  both  may  be  curved. 


13 

RECTIFICATION    OF    CIRCULAR    ARCS. 

For  our  present  purpose  we  have  to  do  only  with  the  rolling  of  a  circle, 
either  upon  its  tangent  or  upon  another  circle,  and  shall  have  frequent 
occasion  to  set  off  upon  a  right  line  a  length  equal  to  that  of  a  given  arc, 
or  upon  a  given  circle  an  arc  equal  in  length  to  a  given  right  line. 

Since  the  circumference  is  3.1416  times  the  diameter,  these  operations 
can  be  performed  arithmetically ;  but  the  following  graphic  process  will  be 
found  equally  accurate  and  much  more  expeditious. 

a 


F     E 


Eig*  ti 

I.  In  Fig.  6,  let  AE  be  tangent  at  A  to  the  given  arc  AB.     Draw 
BA,  produce  it,  making  AG=^AD=y2  chord  AB.     With  center  G  and 
radius  GB,  describe  an  arc  cutting  AE  in  F.     Then  AF=a.rc  AB  (very 
nearly). 

II.  In  Fig.  7,  let  the  given  line  AB,  be  tangent  at  A,  to  the  given 
circle.     Make  AD~}{  AB ;   with   center   D,  and   radius  JDB=i/£  AB, 
describe  an  arc  cutting  the  given  circle  in  E.     Then  arc  AE=AB  (very 
nearly). 

NOTE. — The  arc  thus  rectified  or  found  should  not  measure  over  60°. 
If  the  given  arc  or  line  exceed  this  limit,  it  should  be  bisected. 


14 


The  particular  rolled  curves  to  be  used  are  : 

I.      The  Cycloid,  Fig.  8.     Traced  by  a  point  in  the  circumference  of  a 

/•i 


E 

circle  rolling  upon  its  tangent.  Find  Aa',  the  length  of  a  convenient 
fraction  Aa,  of  the  circumference;  step  this  off  the  required  number  of 
times,  making  ^^"^semi-circumference.  Divide  both  into  the -same 


15 


number  of  equal  parts,  draw  chords  from  P  to  the  points  of  division  on 
the  circle,  with  which,  as  radii,  strike  arcs  about  the  corresponding  points 
on  AE;  the  cycloid  is  tangent  to  all  these  arcs. 

To  find  points  on  the  curve. — When  aC  becomes  contact  radius,  it  has 
the  position  of  a'R,  perpendicular  to  AE.  The  angle  a CP  remaining  un- 
changed, make  a'jRL  equal  to  it:  then  RL  is  the  generating  radius,  and  L 
a  point  on  the  cycloid.  Also  a'L  is  the  normal,  and  a  perpendicular  to  it 
is  tangent  to  the  curve  at  L. 


Fig.  9 


Conversely.     Let  O  be  any  point  on  the  curve  ;  about  this  as  a  center, 


16 


describe  an  arc  with  radius  equal  to  CP,  cutting  CD,  the  path  of  the 
center,  in  S.  Then  OS  is  the  generating  radius;  Stf,  perpendicular  to 
AE,  is  the  contact  radius,  and  b' O  is  normal  to  the  cycloid. 

II.  The  Epicycloid,  Fig.  9.  The  describing  circle  rolls  on  the  outside 
of  another,  whose  center  is  G.  Draw  the  common  tangent  at  A,  set  off  on 
this  the  length  of  Aa  (any  convenient  fraction  of  semi-circumference  AP), 
and  find  the  arc  of  the  base  circle  equal  to  that  length.  Step  this  off  as 
above,  making  ^.fi1— semi-circumference  AP.  The  curve  is  drawfTby 
tangent  arcs  in  the  same  manner  as  the  cycloid.  The  path  of  the  center 
of  the  describing  circle,  is,  in  this  case,  another  circle,  whose  center  is  G, 
and  the  contact  radii  a'JR,  b'S,  are  prolongations  of  the  radii  Ga' ,  Gb' ,  of 
the  base  circle,  which  slightly  modifies  the  processes  of  finding  the  point 
of  the  curve  corresponding  to  a  given  point  of  contact  and  the  converse. 
Fig.  10 


A 


III.      The  Hypocycloid,   Fig.    10.     Traced   by  a  point  in  the  circum- 
ference of  a  circle  rolling  inside  another.     Construction  in  all  respects  the 


17 

same  as  in  the  case  of  the  epicycloid,  and  the  diagrams  being  lettered  to 
correspond  throughout,  no  further  explanation  is  needed. 

In  all  three  of  these  curves,  if  the  rolling  continue  beyond  £,  a  new 
branch  EF  springs  up,  which  is,  of  course,  perfectly  symmetrical  with  EL. 
It  is  to  be  particularly  noted  that  these  branches  are  tangent  to  ED,  and  to 
each  other,  at  E.  These  parts  near  E  are  the  ones  which  require  the 
greatest  care  in  their  construction,  as  they  only  are  employed  in  the  forma- 
tion of  teeth. 


18 


LAYING     OUT     THE     TEETH  —  THE     PITCH     CIRCLE     AND 
CIRCULAR    PITCH. 

If  the  line  of  centers  of  a  pair  of  spur  wheels  be  divided  into  two 
parts  which  are  to  each  other  in  the  same  ratio  as  the  numbers  of  thejjeeth, 
the  circles  of  which  these  parts  are  the  radii  are  called  the  pitch  circles. 
And  the  first  step  in  laying  out  a  pair  of  wheels  is  to  determine  the  fadii 
and  draw  these  circles.  Suppose,  for  example,  that  the  distance  CD, 
between  centers,  in  Fig.  11,  is  given,  and  it  is  required  to  make  two  wheels 

1 


Fig.  11. 

whose  angular  velocities  shall  be  as  2  :  1.  Divide  CD  into  three  equal 
parts,  of  which  AD  is  one,  then  AC  will  measure  two,  and  the  tangent 
circles  shown  are  the  pitch  circles.  Evidently  they  can  move  in  perfect 
rolling  contact  about  their  fixed  centers;  the  linear  motion  AB  is  the 
same,  whether  we  regard  the  point  A,  as  belonging  to  one  circle  or  the 
other.  But  one  will  not  drive  the  other  without  the  possibility  of  slipping, 
which  would  cause  the  velocity  ratio  to  vary ;  hence  the  necessity  of  te«th. 


19 

The  next  step  is  to  divide  each  pitch  circle  into  as  many  equal  parts 
as  its  wheel  is  to  have  teeth.  We  may  give  the  smaller  wheel  any  number 
we  please,  but  the  larger  one  must  have  twice  as  many  in  this  instance. 
The  pitch  of  the  teeth  is  the  length  of  the  circular  arc  obtained  by  this 
subdivision.  Since  the  larger  circumference  is  twice  the  smaller,  but  is 
divided  into  twice  as  many  parts,  the  pitch  arc  is  the  same  in  both  wheels. 
Each  of  these  arcs  must  contain  a  tooth  and  a  space ;  hence  we  may  say 
that  the  pitch  is  the  distance  between  the  centers  or  the  corresponding 
edges  of  rwo  adjacent  teeth,  measured  on  the  pitch  circle,  not  in  a  right  line. 
This  is  sometimes  Jcalled  thejfC/>r///<zr  Pitch,  in  distinction  from  what  is 
known  as  the  Diametral  Pitch,  of  which  hereafter. 

GENERATION    OF    THE    TOOTH     OUTLINE. 

In  Fig.  12,  let  Cand  D  be  the  centers"  of  the  pitch  circles  LM,  RN. 


Tangent  to  these  at  A,  is  a  smaller  circle  whose  center  is  O.  Suppose  all 
the  centers  to  be  fixed,  then  the  three  circles  can  move  in  rolling  contact, 
with  equal  linear  velocities.  Set  off  from  A  the  three  equal  arcs,  AB,  AE, 
AP.  Suppose  a  marking  point  originally  fixed  at  A,  in  the  circumference 
of  the  small  circle ;  then  while  this  travels  to  P,  it  must  trace,  with  refer- 
ence to  £N,  the  curve  BP,  and  with  reference  to  LM,  the  curve  EP.  Now 
the  relative  motions  of  the  circles  are  precisely  the  same  as  though  the 
small  circle  had  rolled  upon  the  outside  of  RN,  and  upon  the  insidtrof 
LM,  regarding  these  as  fixed  base  lines ;  the  curves  are,  therefore,  an  epi- 
cycloid and  a  hypocycloid  respectively;  and  AP  is  their  common  normal. 

If  the  tracing  point  go  on  to  P',  the  arcs  AP',  AE',  AB' ,  being 
equal,  the  resulting  curves  B'P',  E'P',  are  clearly  but  extensions  of  the 
first  pair,  and  AP' ,  is  their  common  normal. 

We  perceive,  then,  that  the  curves  thus  simultaneously  generated  are 
tangent  to  each  other  at  some  point  throughout  the  generation  ;  that  the 
point  of  tangency  is  always  in  the  describing  circle;  and  that  the  com- 
mon normal  always  passes  through  the  fixed  point  A,  upon  the  line  of 
centers. 

Consequently  these  curves  are  correct  outlines  for  parts,  at  least,  of 
teeth;  if  the  curved  lever  CE'  turn,  as  shown  by  the  dotted  arrow,  it  will 
drive  the  other  before  it,  the  point  of  contact  following  the  arc  P'PA, 
until  E'  and  B'  meet  at  A;  and  as  the  common  normal  always  cuts  the 
line  of  centers  at  the  same  point,  the  velocity  ratio  will  be  constant. 

FACE    AND     FLANK. 

The  epicycloid  B'  P',  which  lies  without  its  pitch  circle,  is  called  the 
face;  and  the  hypocycloid  E'P',  which  lies  within  its  pitch  circle,  is  called 
the  flank.  Usually,  each  tooth  has  both  ;  but  wheels  can  be  made;  and 


21 

sometimes  used  to  great  advantage,  in  which  one  of  a  pair  has  faces  only, 
the  other  only  flanks:  we  will  consider  this  case  first. 


SIZE    OF    THE    TOOTH. 

This  depends  upon  the  pitch,  for  the  pitch-arc  must  contain  a  tooth 
and  a  space,  which  might  be  exactly  equal,  were  perfection  in  workmanship 
possible.  Practically,  the  space  must  be  a  little  wider  than  the  tooth;  the 
difference  is  called  back-lash,  and  should  be  made  as  small  as  practicable. 
In  drawing  we  may  disregard  it,  and  make  the  thickness  of  the  tooth  just 
half  the  pitch. 

ARC    AND    ANGLE    OF    ACTION. 

The  angle  through  which  a  wheel  turns,  while  one  of  its  teeth  is  in 
contact  with  a  tooth  of  another  wheel,  is  called  the  angle  of  action,  and  the 
arc  by  which  it  is  measured  is  the  arc  of  action.  This  latter  must  evidently 
be  at  least  equal  to  the  pitch-arc,  in  order  that  each  tooth  may  continue  in 
gear  until  the  next  one  begins  to  act,  and  it  should  be  considerably  greater. 


A     PAIR    OF    WHEELS — LIMITING     CASE. 

In  Fig.  13,  the  pitch  and  describing  circle  being  drawn  as  in  Fig.  12, 
let  AB,  AE,  be  the  pitch  arcs,  and  AP  an  equal  arc  on  the  describing 
circle.  Then  the  face  for  the  tooth  of  RN,  can  not  be  less  than  BP,  since, 
if  made,  as  shown,  of  exactly  that  height,  contact  is  ending  at  P,  at  the 
very  instant  the  next  tooth  begins  to  act  at  A.  Bisect  AB  in  H,  which 
gives  the  thickness  of  the  tooth  ;  and  draw  through  H,  a  reversed  face  sim- 
ilar to  JBP.  The  conditions  are  purposely  so  chosen  that  this  second  face 


Fly.  13 


passes  through  P.  The  case  is,  therefore,  a  barely  possible  one  ;  the  tooth 
is  pointed,  and  just  high  enough  to  continue  in  gear  until  the  next  one 
begins  to  act.  We  found  that  the  face  must  be  of  the  height  BP,  in  order 
to  secure  this  arc  of  action ;  drawing  PD,  which  cuts  the  pitch  circle  in  G, 
we  find  in  this  case  that  BG  is  just  half  the  thickness  of  the  tooth.  Had 
it  been  greater,  GH  must  have  been  less,  so  that  the  face  through  Ht  would 
not  have  passed  through  P,  but  between  P  and  G,  and  the  case  would 
have  been  impracticable ;  it  would  then  have  been  necessary  to  reduce  the 
pitch  and  give  both  wheels  more  teeth.  But  if  BG  had  been  less  than  half 
the  thickness  of  the  tooth,  we  could  either  make  the  tooth  higher,  or  give 
it  some  thickness  at  the  top,  as  in  Fig.  14. 


o  •  Is 

The  acting  flank  is  EP ';  but  in  order  to  let  the  teeth  of  the  other 
wheel  pass,  the  hypocycloid  is  extended  to  /,  making  the  depth  of  the 
space  a  little  greater  than  PG;  the  difference  is  called  clearance,  and  a 
similar  provision  is  made  in  the  other  wheel  by  cutting  in  radially,  as 
shown*  at  A,  H,  JB,  a  little  below  the  pitch  circle.  The  tooth  of  LM,  is 
completed  by  bisecting  the  pitch-arc  AE,  at  F,  and  drawing  the  curves 
AK',  FJ,  etc.,  similar  to  El. 

A     PRACTICAL    CASE. 

Limiting  cases  like  the  preceding  are  to  be  avoided  in  practice.  A 
pointed  tooth  is  bad,  as  being  weak  and  liable  to  wear  at  the  top,  and 


24 

even  if  it  be  not  pointed,  the  angle  of  action  should  be  greater,  as  other- 
wise, the  least  wear  at  the  top  reduces  the  face  below  the  requisite  height, 
which  affects  the  velocity  ratio.  A  reasonable  case  is  shown  in  Fig.  14 ; 
the  arc  of  action  is  \y2  times  the  pitch,  and  drawing  the  radial  line  PS,  we 
find  BG  much  less  than  y2  BH,  thus  giving  the  tooth  a  thickness  PK,  at 
the  top. 

APPROACHING    AND    RECEDING    ACTION. 

In  Figs.  13  and  14,  the  action  takes  place  wholly  on  one  side  of  the 
line  of  centers.  If  filVbe  the  driver  (the  direction  being  as  shown  by  the 
arrows)  the  action  begins  at  A  and  ends  at  P,  the  point  of  contact  contin- 
ually receding  from  the  line  of  centers ;  in  which  case  AJ3,  AE,  are  called 
arcs  of  recess,  or  of  receding  action.  If  LM drive  (in  the  opposite  direc- 
tion) the  action  begins  at  Pending  at  A;  the  point  of  contact  is  always 
approaching  the  line  of  centers,  and  AB,  AE,  are  then  called  arcs  of  ap- 
proach, or  of  approaching  action. 

It  has  been  found  by  experience  that  the  friction  is  greater  and  more 
injurious  in  the  latter  case  than  in  the  former ;  hence  when  such  wheels  are 
used,  the  one  with  faces  only  should  always  drive.  But  even  then,  there  is 
one  drawback,  which  will  be  seen  by  reference  to  Fig.  12.  The  longer  the 
arc  of  action,  the  longer  the  face  of  the  tooth,  and  the  greater  the 
obliquity  of  the  line  of  action,  that  is,  its  inclination  to  the  common  tangent 
TT,  of  the  pitch  circles.  The  pressure  as  well  as  the  motion  is  transmitted 
in  this  line,  and  the  greater  its  inclination  to  TT,  the  greater  will  be  the 
component  of  pressure  in  the  line  of  centers,  tending  to  cause  friction  in 
the  bearings. 

Consequently  such  wheels  are  better  suited  for  use  in  light  mechanism 
where  the  teeth  can  be  made  small  and  numerous,  and  smoothness  of  action 
is  important,  than  for  the  transmission  of  heavy  pressure. 


25 


TEETH    WITH     BOTH    FACES    AND     FLANKS. 

By  giving  faces  and  flanks  to  the  teeth  of  each  wheel  of  a  pair,  we 
can  secure  a  given  angle  of  action  with  shorter  faces,  consequently  with 
less  sliding  and  less  obliquity  of  action.  Also,  the  action  will  take  place 
partly  before,  and  partly  after,  the  point  of  contact  reaches  the  line  of 
centers.  If  a  wheel  is  both  to  drive  and  to  follow,  the  arcs  of  approach 


.  15 


R 


and  recess  may  be  made  equal;  but  if  one  of  a  pair  is  always  the  driver, 
it  may  be  desirable  to  make  the  arc  of  recess  the  greater,  in  order  to  reduce 
the  amount  of  the  more  injurious  friction. 

The  construction  is  shown  in  Fig.  15 ;  all  that  relates  to  the  face  BP 


for  RN,  and  the  flank  EP  for  LM,  is  precisely  the  same  as  in  Fig.  13,  and 
the  lettering  being  so  far  made  to  correspond,  no  further  explanation  is 
needed.  To 'complete  the  teeth,  another  describing  circle  is  used,  on  the 
opposite  side  of  the  pitch  circles,  which  generates  the  face  OF  tor  LM,  and 
flank  OK  for  RN.  If  we  assume  the  arc  of  action  on  that  side  of  CD,  as 
AFvr  AK,  the  possibility  of  securing  it  with  a  given  number  of  teeth  is 
at  once  ascertained  by  making  the  arc  AO  equal  to  Af,  and  drawing  OC, 
cutting  LM  in  /.-  if  FI  be  less  than  half  the  thickness  of  the  tooth- as 
required  by  the  pitch,  or  equal  to  it,  the  construction  is  possible,  the  tooth 
in  the  latter  event  being  pointed ;  if  greater  it  is  impracticable.  If  it  be 
found  feasible,  we  have  only  to  draw  the  epicycloid  OF,  which  joined  to 
EP  completes  the  outline  of  the  tooth  for  LM,  and  the  hypocycloid  OK, 
joining  it  to  BP,  which  finishes  the  outline  of  the  tooth  of  RN.  That  is 
to  say,  these  are  the  whole  of  the  acting  outlines ;  the  flanks  must  be  "ex- 
tended to  a  greater  depth  in  order  to  give  clearance,  as  already  explained. 

The  operation  will  be  readily  seen ;  as  in  the  diagram  the  acting  side 
of  a  tooth  of  each  wheel  is  drawn  in  two  positions.  Supposing  RJV  to 
drive,  the  action  begins  at  O,  the  driver's  flank  pushing  the  face  of  the 
follower,  and  the  point  of  contact  moving  in  the  arc  OA,  until  K  and  F 
meet  at  A. 

The  face  of  the  driver  then  urges  the  follower's  flank,  the  point  of 
contact  now  traveling  in  the  arc  OP,  and  at  P  the  action  ends. 

We  see,  then,  that  the  angle  of  approach  depends  upon  the  length  of 
the  follower's  face,  and  the  angle  of  recess  upon  that  of  the  driver's  face ; 
and  if  these  lengths  be  assumed  or  given,  those  angles  are  readily  found, — 
as  for  instance,  had  the  length  FO  been  assigned,  it  is  only  necessary  to 
strike  an  arc  about  C  with  radius  CO,  which,  cutting  the  describing  circle 
in  O,  gives  OA  the  length  of  the  arc  of  approach,  which  is  then  to  be  set 
off  on  ZJ/and  RN,  as  AF,  AK. 


27 

A  Practicable  Example. 

The  diagram  Fig.  15,  is  drawn  without  regard  to   practical  propor- 
tions, in  order  to  make  the  construction  clear;  but  in  Fig.  16  we  have 


I) 


shown   a   feasible  case ;    the  cut  is  half  size, '  and   the   conditions  are  as 
follows  : 

Distance  between  centers,  27  inches.     Wheels  to  have  63  and  45  teeth, 

the  smaller  to  drive. 

Angle  of  action  to  be  2f  times  the  pitch. 
Angle  of  recess  to  be  one-third  greater  than  the  angle  of  approach. 


We  have,  then,  63:45::7:5,  7+5=12,  tf=2J,  2Jx7=15f,  2Jx5=ll#,  for 
the  radii  of  the  pitch  circles.  And  2|=V,  which  is  to  be  divided  into  parts 
in  the  ratio  of  3:4;  whence,  3+4=7,  Y-=-7=f,  f  X3=li  times  the  pitch 

=angle  of  approach, 
£X4=1£  times  the  pitch 
=angle  of  recess. 

INTERCHANGEABLE   WHEELS. 

Inasmuch  as  the  face  and  the  flank  which  act  in  contact  are  generated 
by  the  same  describing  circle,  it  makes  no  difference  whether  the  diameter 
of  the  one  which  traces  the  other  face  and  flank  be  the  same  or  not,  in 
laying  out  a  single  pair  of  wheels;  and  in  Fig.  15  the  describing  circles  are 
of  different  diameters.  But  for  the  very  reason  just  stated,  it  is  clear  that 
if  we  wish  to  make  a  number  of  wheels,  any  one  of  which  will  gear  with 
any  other  one,  we  must  use  the  same  describing  circle  for  all  the  faces  and 
all  the  flanks. 

SIZE    OF    THE    DESCRIBING    CIRCLE. 

In  making  such  a  set  of  wheels,  the  question  at  once  arises,  how  large 
shall  the  describing  circle  be?  This  depends  upon  a  property  of  the 
hypocycloid,  illustrated  by  Figs.  17  and  18.  In  Fig.  17,  the  describing 
circle  is  half  the  size  of  the  base  circle,  and  the  line  traced  by  the  point  A 
is  merely  the  diameter  AD.  For  after  rolling  till  the  point  of  contact  is 
B  (taken  at  pleasure),  the  describing  circle  cuts  AD  in  P,  its  center  mean- 
time going  from  E  to  F.  Draw  BFC,  and  PF;  then  the  angle  A  CB  is 
half  the  angle  PFB,  but  the  radius  A  C  is  twice  the  radius  BF,  therefore 
the  arcs  AB,  PJB,  are  equal.  In  Fig.  18,  the  point  P  will  trace  the  curve 


Fig.  18 


AS,  by  rolling  in  one  direction,  if  we  regard  it  as  in  the  circumference  of 
the  smaller  circle ;  but  if  the  same  point  be  carried  by  the  larger  describ- 
ing circle,  it  will  trace  the  same  curve  by  rolling  in  the  other  direction. 
If,  then,  in  any  case  the  describing  circle  be  half  the  size  of  the  pitch  circle, 
the  flanks  will  be  radial,  as  in  Fig.  19 ;  if  it  be  less,  they  will  spread  out 
Fly.  19  Fig.  20  Fig.  21 


30 

toward  the  root  of  the  tooth,  giving  a  stronger  form,  as  in  Fig.  20 ;  but  if 
greater,  the  flanks  will  curve  in  toward  each  other,  as  in  Fig.  21,  whereby 
the  teeth  become  weaker  and  difficult  to  make. 

From  this,  the  safe  practical  deduction  is,  that  the  describing  circle  for 
a  set  of  wheels  should  not  be  more  than  half  the  diameter  of  the  smallest 
wheel ;  and  in  laying  out  a  single  pair,  two  describing  circles,  each  of  I 
the  diameter  of  its  pitch  circle,  give  good  practical  forms  to  the  teeth^  for 
general  purposes. 

Still,  the  face  is  shorter,  and  the  obliquity  of  action  less,  for  a  given 
arc  of  action,  the  larger  the  describing  circle ;  so  that  for  very  delicate 
mechanism,  it  is  possible  that  the  gain  from  these  causes  would  sometimes 
render  it  advisable  to  use  teeth  of  the  form  shown  in  Fig.  21.  They  may 
be  much  strengthened  by  the  use  of  large  fillets  at  the  junction  of  the^side 
and  bottom  of  the  space,  which  is  quite  admissible,  since  the  acting  depth 
of  the  flank  is  comparatively  small,  as  has  been  shown. 

RACK    AND    WHEEL. 

A  rack  is  simply  an  infinitely  large  wheel.  The  curvature  of  a  circle 
diminishes  as  the  radius  increases,  and  disappears  when  the  radius  becomes 
infinite ;  so  that  the  pitch  line  of  a  rack  is  only  a  straight  tangent  to  the 
pitch  circle  of  the  wheel  with  which  it  works,  and  the  line  of  centers  become 
a  perpendicular  to  this  pitch  line,  through  the  center  of  the  wheel. 

The  rack  will  travel  through  a  distance  equal  to  the  circumference  of 
the  pitch  circle  of  the  wheel  during  one  revolution  of  the  latter,  whatever 
the  number  of  teeth,  and  in  the  same  proportion  for  any  fraction  of  a  revo- 
lution. The  pitch  of  the  rack  teeth,  therefore,  is  found  by  rectifying  the 
pitch  arc  of  the  wheel,  whatever  that  may  be,  and  setting  off  that  length 
upon  the  pitch  line. 


31 

The  construction  is  shown  in  Fig.  22.  We  have  here  shown  the  two 
describing  circles  as  of  the  same  size,  and  it  is  clear  that  if  the  same  circle 
be  used  to  generate  the  faces  and  flanks  of  a  -set  of  wheels,  any  one  of 
them  will  gear  with  the  rack  if  the  pitch  be  also  the  same. 


Fig.  22 


Evidently  both  faces  and  flanks  of  the  rack  teeth  are  cycloids,  being 
generated  by  the  rolling  of  a  circle  upon  the  pitch  line.  If  the  length  BP 
of  the  face,  be  assumed  or  given,  a  line  parallel  to  RN  cute  the  generating 
circle  in  P,  thus  determining  AP,  to  which  AB  must  be  made  equal,  and 
fixing  the  part  of  the  action  which  will  take  place  on  the  right  of  CD.  Or 
if  AB  be  assigned,  we  make  AP  equal  to  it,  thus  ascertaining  the  necessary 
length  of  face.  In  either  case,  PS  is  now  to  be  drawn  perpendicular  to 


32 

the  pitch  line,  which  it  cuts  at  G,  and  as  in  the  preceding  constructions 
BG  cannot  be  greater,  and  should  be  less,  than  half  the  thickness  of  the 
tooth  as  determined  by  the  pitch.  The  part  of  the  action,  which  will  take 
place  on  the  left  of  CD,  depends  upon  the  length  of  the  face  of  the  wheel 
tooth,  and  is  ascertained  as  in  the  cases  previously  explained. 

ARBITRARY     PROPORTIONS. 

It  is  not  necessary  in  all  cases,  indeed  probably  not  in  the  majority  of 
cases,  to  pay  particular  attention  to  the  relative  amounts  of  the  approach- 
ing and  receding  action.  It  is  a  very  common  practice  to  make  the 
whole  height  of  the  tooth  a  certain  fraction  of  the  pitch ;  the  part  which 
projects  outside  the  pitch  circle  being  made  a  little  less  than  that  within, 
by  which  the  clearance  is  provided  for.  Thus,  in  Fig.  23,  the  whole  height 


Fig.  23. 

/,  is  |  of  the  pitch  ;  and  the  part  h  is  to  d  as  11  :  13. 
tions  which  have  been  extensively  used  are  as  follows : 
/=§  pitch ;  h  :  d ::  4  :  5. 
"      h  :  d ::  3  :  4. 


Two  other  propor- 


33 

Teeth  proportioned  according  to  either  of  these  rules  will  work  satis- 
factorily for  most  purposes,  if  there  be  at  least  12  teeth  upon  the  smallest 
wheel.  But  if  occasion  arises,  as  it  may,  for  the  use  of  a  pinion  with  only 
six  or  eight  "leaves,"  (as  the  teeth  of  very  small  wheels  are  sometimes 
called,)  it  will  be  found  advantageous,  if  not  absolutely  necessary,  to  make 
the  faces  of  those  leaves  longer.  As  for  the  angle  of  action,  and  the  pro- 
portion of  the  angle  of  approach  to  that  of  recess,  both  these  things  will, 
of  course,  vary  according  to  the  numbers  of  teeth,  whichever  of  these 
systems  be  adopted.  It  may  be  added,  that  in  connection  with  these  rules, 
instructions  are  frequently  given  as  to  the  amount  of  back-lash,  which  is 
also,  according  to  them,  a  definite  fraction  of  the  pitch.  This  is,  no  doubt, 
very  proper  in  reference  to  wheels  cast  from  patterns,  but  there  certainly 
does  not  seem  to  be  any  reason  for  it  if  the  teeth  are  to  be  cut ;  for  whatever 
the  pitch,  it  is  practically  sufficient  that  the  backs  of  the  teeth  should  barely 
clear  each  other  when  the  fronts  are  in  driving  contact. 

DIAMETRAL    PITCH. 

In  designing  spur  gearing  it  is  necessary  to  find  the  circular  pitch,  not 
only  because,  as  we  have  seen,  it  is  used  in  the  graphic  construction,  but 
because  the  strength  of  the  tooth  depends  upon  its  thickness.  Were  there 
nothing  to  the  contrary,  it  would  be  most  convenient  to  express  the  pitch 
in  whole  numbers  or  manageable  fractions,  as  2  inch  pitch,  f  inch  pitch, 
and  so  on.  But  as  the  circumference  is  3.1416  times  the  diameter,  awk- 
ward decimals  will  often  appear  in  the  values  of  the  diameters  of  the  pitch 
circles,  if  this  plan  be  adhered  to.  Now,  if  the  tooth  be  strong  enough,  it 
matters  not  if  it  be  a  little  stronger ;  and  it  is  practically  much  more  im- 
portant to  have  the  diameter  a  whole  number,  or  a  convenient  fraction, 
than  that  the  circular  pitch  should  be  either  the  one  or  the  other.  This  is 


34 

accomplished  by  the  use  of  what  is  called  the  diametral pitch ;  which  is 
simply  the  quotient  found  by  dividing  the  diameter  of  the  pitch  circle, 
instead  of  the  circumference,  into  as  many  equal  parts  as  the  wheel  has 
teeth. 

Whence,    Diametral   Pitch=     Diameter        and    circular   Pitch=Di- 
No.  of  Teeth. 

ametral  Pitch  X  3.1416. 

In  the  use  of  this  system,  convenient  values  of  the  diametral  pitch  are 
selected,  each  being  a  fraction  with  unity  for  its  numerator  and  an  integer 
for  its  denominator,  as  1,  *,  i,  J-,  TV,  TJ,  etc. 

The  denominators  of  these  fractions  only  are  commonly  used  in  giv- 
ing the  diametral  pitch;  thus,  an  "8-pitch  wheel"  is  one  which  has  eight 
teeth  for  each  inch  of  diameter,  or  whose  diametral  pitch  is  \" .  -This  is, 

in  fact,  merely  inverting  the  fraction,  and  giving  the  value  of  — ; 

Diameter 

thus,  let  a  wheel  of  16  inches  diameter  have  80  teeth;  then,  $$=\"  = 
diametral  pitch,  but  |£  =  5,  and  we  call  it  a  "5-pitch"  wheel.  By  this 
system  the  calculations  as  to  diameter  and  number  of  teeth  are  made  very 
simple  ;  as,  for  example  : 

Required,  the  diameter  of  a  4-pitch  wheel  with  37  teeth : 

V  =  9i  —  diameter. 

How  many  teeth  of  16-pitch  on  a  wheel  of  3|  diameter  ? 
3|  X  16  =  62  =  No.  of  teeth. 

The  tooth  may  be  made  to  project  outside  the  pitch  circle,  a  definite 
fraction  of  the  diametral  pitch,  as  in  the  case  of  the  circular  pitch;  and 
thus  the  size  of  the  blank  may  be  readily  ascertained.  If  this  projection  be 


35 

made,  for  instance,  1 J  times  the  diametral  pitch,  the  face  of  the  tooth  will 
be  nearly  as  long  as  that  found  by  the  first  of  the  arbitrary  rules  above 
given ;  and  the  diameter  of  the  blank  is  determined  by  simply  adding  to 
that  of  the  pitch  circle,  2}  times  the  diametral  pitch. 


PART  II. 


THE  MANUFACTURE  OF  ACCURATE  GEAR  CUTTERST* 

In  cutting  a  spur  wheel  it  is  essential  that  the  contour  of  the  milling 
cutter  conform  precisely  to  that  of  the  space  between  two  teeth.  The 
obvious  disadvantages  of  turning  the  cutter  by  hand  to  fit  a  template  filed 
out,  if  not  laid  out,  by  hand,  have  already  been  alluded  to  ;  also  the  fact 
that  they  have  been  in  part  avoided,  by  the  use  of  mechanical  mearis^for 
describing  the  required  curves  on  the  template. 

This  is,  however,  but  one  step,  and  that  of  the  shortest;  for  by  the 
methods  previously  explained,  the  epicycloidal  curves  can  be  drawn  on 
metal  with  comparative  speed  and  extreme  accuracy.  There  yet  remain  the 
laborious  processes  of  making  the  template,  and  making  the  cutter  fit  it 
when  made.  When  these  things  are  done  by  hand,  exact  duplication  of 
templates,  cutters  or  wheels,  is  a  matter  of  impossibility,  to  all  practical 
intents  and  purposes. 

But  this  becomes,  on  the  contrary,  a  matter  of  ease  and  perfect  cer- 
tainty, when  the  work  is  done  by  the  two  machines  of  which  we  give 
illustrations.  They  were  recently  introduced  by  The  Pratt  and  Whitney 
Company,  in  whose  shops  they  may  be  seen  in  successful  operation ;  and 
they  are  well  worthy  of  study,  not  only  on  account  of  the  ingenuity  and 
beauty  of  their  movements,  but  as  models  of  skillful  planning  and  rare 
examples  of  the  practical  embodiment  of  correct  theory. 


,: 


o 

a 

w 


w 


71 
bi) 


THE    EPICYCLOIDAL    MILLING    ENGINE. 


The  object  of  this  machine  is  to  form  the  template  subsequently  used 
as  a  guide  in  shaping  the  cutter.  Its  general  appearance  is  shown  in  Fig. 
1 ;  the  operating  parts  are  more  distinctly  shown  in  Fig.  2,  which  is  a 
view  taken  nearly  from  above ;  the  action  is  illustrated  in  Figs.  3,  4,  5, 

6,  7,  8. 


In  Fig.  3,  A,A,  is  a  portion  of  a  flat  ring,  fixed  to  the  framing ;  this 
represents  a  pitch  circle.     B,  is  a  disc,  representing  the  describing  circle  ; 


40 

this  turns  freely  upon  a  tubular  stud  E,  fixed  in  the  carrier  C,  which  turns 
about  a  pivot  D,  fixed  to  the  frame  at  the  center  of  A.  By  means  of  the 
clamped  socket,  capable  of  sliding  upon  the  rod,  the  position  of  D  may 
be  adjusted  to  suit  the  radius  of  A.  Thus  as  C  moves,  the  disc  can  roll 
upon  the  edge  of  A,  and  is  compelled  to  do  so  by  the  flexible  steel  ribbon 
shown  by  the  heavy  line,  which  is  wrapped  round  and  secured  to  both 
pieces,  due  allowance  for  its  thickness  being  made  in  adjusting  their  radii. 


E'  is  a  second  tubular  stud  fixed  in  the  carrier,  at  the  same  distance  from 
the  pitch  circle  as  the  other,  but  on  the  opposite  side ;  the  centers  of~the 
two  studs  lying  on  a  right  line  through  D.  Upon  these  two  studs  turn  the 


41 

two  worm  wheels,  F,  f,  shown  in  Fig.  4 ;  these  are  in  a  plane  above  A 
and  B,  so  that  the  axis  of  the  worm  G,  is  vertically  over  the  common 
tangent  of  the  pitch  and  describing  circles;  the  relative  positions  of  these 
and  other  parts  will  be  most  clearly  seen  by  a  study  of  the  vertical  section, 
Fig.  8.  The  worm  G,  is  supported  in  bearings  secured  to  the  carrier  C, 
and  is  driven  by  another  small  worm  turned  by  the  pulley  /,  as  seen  in 
Fig.  2 ;  the  driving  cord,  passing  through  suitable  guiding  pulleys,  is  kept 
at  uniform  tension  by  a  weight,  however  C  moves ;  this  is  shown  in  Figs. 
1  and  2. 

Upon  the  same  studs,  in  a  plane  still  higher  than  the  worm-wheels, 
turn  the  two  discs  H,  H' ,  Figs.  5,  6,  7.  The  diameters  of  these  are  equal, 
and  precisely  the  same  as  those  of  the  describing  circles  which  they 
represent,  with  due  allowance,  again,  for  the  thickness  of  a  steel  ribbon, 
by  which  these,  also,  are  connected.  It  will  be  understood  that  each  of 
these  discs  is  secured  to  the  worm-wheel  below  it,  and  the  outer  one  of 
these  to  the  disc  B,  so  that  as  the  worm  G  turns,  H  and  H'  are  rotated  in 
opposite  directions,  the  motion  of  H  being  identical  with  that  of  B ;  this 
last  is  a  rolling  one  upon  the  edge  of  A,  the  carrier  Cwith  all  its  attached 
mechanism  moving  around  D  at  the  same  time.  Ultimately,  then,  the 
motions  of  H,  H' ,  are  those  of  two  equal  describing  circles  rolling  in  ex- 
ternal and  internal  contact  with  a  fixed  pitch  circle. 

In  the  edge  of  each  disc  a  semi-circular  recess  is  formed,  into  which  is 
accurately  fitted  a  cylinder  /,  provided  with  flanges,  between  which  the 
discs  fit  so  as  to  prevent  end  play.  This  cylinder  is  perforated  for  the 
passage  of  the  steel  ribbon,  the  sides  of  the  opening,  as  shown  in  Fig.  5, 
having  the  same  curvature  as  the  rims  of  the  discs.  Thus  when  these 
recesses  are  opposite  each  other,  as  in-Fig.  6,  the  cylinder  J  fills  them  both, 
and  the  tendency  of  the  steel  ribbon  is  to  carry  it  along  with  H  when  C 


42 

moves  to  one  side  of  this  position,  as  in  Fig.  7,  and  along  with  H'  when  C 
moves  to  the  other  side,  as  in  Fig.  5. 

This  action  is  made  positively  certain  by  means  of  the  hooks  K  K' , 
which  catch  into  recesses  formed  in  the  upper  flange  of  J,  as  seen  in  Fig.v 
6.  The  spindles,  with  which  these  hooks  turn,  extend  through  the  hollow 
studs,  and  the  coiled  springs  attached,  to  their  lower  ends,  as  seen  in  Fig.  8, 
urge  the  hooks  in  the  directions  of  their  points  ;  their  motions  being 
limited  by  stops  o,  o',  fixed,  not  in  the  discs  H,  H' ,  but  in  projecti«g*-ct>llars 
on  the  upper  ends  of  the  tubular  studs.  The  action  will  be  readily  traced 
by  comparing  Fig.  6  with  Fig.  7 ;  as  C  goes  to  the  left,  the  hook  K'  is  left 


behind,  but  the  other  one,  K,  cannot  escape  from  its  engagement  with  the 
flange  of  J;  which,  accordingly,  is  carried  along  with  H  by  the  combined 
action  of  the  hook  and  the  steel  ribbon. 

On  the  top  of  the  upper  flange  of  /,  is  secured  a  bracket,  carrying  the 
bearing  of  a  vertical  spindle  Z,  whose  center  line  is  a  prolongatioivof  that 


43 

of  J  itself.  This  spindle  is  driven  by  the  spur  wheel  N,  keyed  on  its 
upper  end,  through  a  flexible  train  of  gearing  seen  in  Fig.  2  :  at  its  lower 
end  it  carries  a  small  milling  cutter  M,  which  shapes  the  edge  of  the  tem- 
plate T,  Fig.  7,  firmly  clamped  to  the  framing. 

When  the  machine  is  in  operation,  a  heavy  weight  seen  in  Fig.  1,  acts 
to  move  C  about  the  pivot  D,  being  attached  to  the  carrier  by  a  cord  guided 
by  suitably  arranged  pulleys  ;  this  keeps  the  cutter  Mup  to  its  work,  while 
the  spindle  L  is  independently  driven,  and  the  duty  left  for  the  worm  G  to 
perform,  is  merely  that  of  controlling  the  motions  of  the  cutter  by  the 
means  above  described,  and  regulating  their  speed. 

The  center  line  of  the  cutter  is  thus  automatically  compelled  to  travel 
in  the  path  RS,  Fig.  7,  composed  of  an  epicycloid  and  a  hypocycloid  if  A  A 
be  a  segment  of  a  circle  as  here  shown ;  or  of  two  cycloids,  if  A  A  be  a 
straight  bar.  The  radius  of  the  cutter  being  constant,  the  edge  of  the  tem- 
plate Tis  cut  to  an  outline  also  composed  of  two  curves  ;  since  the  radius 
M  is  small,  this  outline  closely  resembles  RS,  but  particular  attention  is 
called  to  the  fact  that  it  is  not  identical  with  it,  nor  yet  composed  of  truly 
epicycloidal  curves  of  any  generation  whatever :  the  result  of  which  will  be 
subsequently  explained. 

NUMBER    AND     SIZES    OF    TEMPLATES. 

With  a  given  pitch,  every  additional  tooth  increases  the  diameter  of 
the  wheel,  and  changes  the  form  of  the  epicycloid  j  so  that  it  would  ap- 
pear necessary  to  have  as  many  different  cutters,  as  there  are  wheels  to  be 
made,  of  any  one  pitch. 

But  the  proportional  increment,  and  the  actual  change  of  form,  due  to 
the  addition  of  one  tooth,  becomes  less  as  the  wheel  becomes  larger ;  and 
the  alteration  in  the  outline  soon  becomes  imperceptible.  Going  still  far- 


44 

ther,  we  can  presently  add  more  teeth  without  producing  a  sensible  varia- 
tion in  the  contour.  That  is  to  say,  several  wheels  can  be  cut  with  the 
same  cutter,  without  introducing  a  perceptible  error.  It  is  obvions  that 
this  variation  in  the  form,  is  least  near  the  pitch  circle,  which  is  the  only 
part  of  the  epicycloid  made  use  of;  and  Prof.  Willis  many  years  ago 
deduced  theoretically,  what  has  since  been  abundantly  proved  by  practice, 
that  instead  of  an  infinite  number  of  cutters,  24  are  sufficient  of  one  pitch, 
for  making  all  wheels,  from  one  with  12  teeth  up  to  a  rack. 


Accordingly,  in  using  the  epicycloidal  milling  engine,  for  forming  the 
template,  segments  of  pitch  circles  are  provided  of  the  following  diameters 
(in  inches) : 

12,  16,  20,  27,  43,  100, 

13,  17,  21,  30,  50,  150, 

14,  18,  23,  34,  60,  300, 

15,  19,  25,  38,  75,  oo.    — 


45 

The  diameter  of  the  discs  which  act  as  describing  circles,  is  7£  inches, 
and  that  of  the  milling  cutter  which  shapes  the  edge  of  the  template,  is  I 
of  an  inch. 

Now  if  we  make  a  set  of  1 -pitch  wheels  with  the  diameters  above 
given,  the  smallest  will  have  twelve  teeth,  and  the  one  with  fifteen  teeth 
will  have  radial  flanks.  The  curves  will  be  the  same  whatever  the  pitch ; 
but  as  shown  in  Fig.  9,  the  blank  should  be  adjusted  in  the  epicycloidal 
engine,  so  that  its  lower  edge  shall  be  iVh  of  an  inch  (the  radius  of  the 
cutter  Af)  above  the  bottom  of  the  space ;  also  its  relation  to  the  side  of 
the  proposed  tooth  should  be  as  here  shown.  As  previously  explained, 
the  depth  of  the  space  depends  upon  the  pitch.  In  the  system 
adopted  by  The  Pratt  &  Whitney  Company,  the  whole  height  of  the  tooth 
is  2^  times  the  diametral  pitch,  the  projection  outside  the  pitch  circle  being 
just  equal  to  the  pitch,  so  that  diameter  of  blank  =  diameter  of  pitch  circle 
+  2  X  diametral  pitch. 

We  have  now  to  show  how,  from  a  single  set  of  what  may  be  called 
1-pitch  templates,  complete  sets  of  cutters  of  the  true  epicycloidal  contour 
may  be  made  of  the  same  or  any  less  pitch. 

THE    PANTAGRAPHIC    ENGINE    FOR    FORMING    CUTTERS. 

In  Fig.  9,  the  edge  TT,  is  shaped  by  the  cutter  M,  whose  center 
travels  in  the  path  J?S,  therefore  these  two  lines  are  at  a  constant  normal 
distance  from  each  other.  Let  a  roller  P,  of  any  reasonable  diameter,  be 
run  along  TT,  its  center  will  trace  the  line  UV,  which  is  at  a  constant 
normal  distance  from  TT,  and  therefore  from  RS.  Let  the  normal  distance 
between  £7Fand  fiSbe  the  radius  of  another  milling  cutter  N,  having  the 
same  axis  as  the  roller  P,  and  carried  by  it,  but  in  a  different  plane,  as 
shown  in  the  side  view  ;  then  whatever  ^Vcuts  will  have  RS  for  its  contour, 
if  it  lie  upon  the  same  side  of  the  cutter  as  the  template. 


46 


Now  if  TT  be  a  1-pitch  template  as  above  mentioned,  it  is  clear  that 
^Vwill  correctly  shape  a  cutting  edge  of  a  gear  cutter  for  a  1-pitch  wheel. 
The  same  figure,  reduced  to  half  size,  would  correctly  represent  the  for- 
mation of  a  cutter  for  a  2-pitch  wheel  of  the  same  number  of  teeth;  if  to 
quarter  size,  that  of  a  cutter  for  a  4-pitch  wheel,  and  so  on. 

But  since  the  actual  size  and  curvature  of  the  contour  thus  determined, 
depend  upon  the  dimensions  and  motion  of  the  cutter  N,  it  will  be  seen 
that  the  same  result  will  practically  be  accomplished,  if  these,  onlyj-be  re- 
duced ;  the  size  of  the  template,  the  diameter  and  the  path  of  the  roller 
remaining  unchanged. 


Fig  10 


The  nature  of  the  means  by  which  this  is  effected  in  the  Pantagraphic 
Engine,  is  illustrated  in  Fig.  10.  The  milling  cutter  N,  is  driven  by 
a  flexible  train  acting  upon  the  wheel  O;  its  spindle  is  carried  -by  the 
bracket  JB,  which  can  slide  from  right  to.  left  upon  the  piece  A,  and  this, 


47 

again,  is  free  to  slide  in  the  framed.  These  two  motions  are  in  horizontal 
planes,  and  perpendicular  to  each  other. 

The  upper  end  of  the  long  lever  PC,  is  formed  into  a  ball,  working 
in  a  socket  which  is  fixed  to  B.  Over  the  cylindrical  upper  part  of  this 
lever  slides  an  accurately  fitted  sleeve  D,  partly  spherical  externally,  and 
working  in  a  socket  which  can  be  clamped  at  any  height  on  the  frame  f. 
The  lower  end  P,  of  this  lever  being  accurately  turned,  corresponds  to  the 
roller  P  in  Fig.  9,  and  is  moved  along  the  edge  of  the  template  T,  which 
is  fastened  in  the  frame  in  an  invariable  position. 

By  clamping  D  at  various  heights,  the  ratio  of  the  lever  arms  PD, 
DC,  may  be  varied  at  will,  and  the  axis  of  N  made  to  travel  in  a  path 
similar  to  that  of  the  axis  of  P,  but  as  many  times  smaller  as  we  choose ; 
and  the  diameter  of  TV  is  made  less  than  that  of  P  in  the  same  proportion. 

The  template  being  on  the  left  of  the  roller,  the  cutter  to  be  shaped  is 
placed  on  the  right  of  N,  as  shown  in  the  plan  view  at  Z,  because  the  lever 
reverses  the  movement. 

This  arrangement  is  not  mathematically  perfect,  by  reason  of  the 
angular  vibration  of  the  lever.  This  is,  however,  very  small,  owing  to 
the  length  of  the  lever;  it  might  have  been  compensated  for  by  the  intro- 
duction of  another  universal  joint,  which  would  practically  have  introduced 
an  error  greater  than  the  one  to  be  obviated,  and  it  has,  with  good  judg- 
ment, been  omitted. 

The  gear  cutter  is  turned  nearly  to  the  required  form,  the  notches  are 
cut  in  it,  and  the  duty  of  the  pantagraphic  engine  is  merely  to  give  the 
finishing  touch  to  each  cutting  edge,  and  give  it  the  correct  outline.  It  is 
obvious  that  this  machine  is  in  no  way  connected  with,  or  dependent  upon, 
the  epicycloidal  engine;  but  by  the  use  of  proper  templates  it  will  make 
cutters  for  any  desired  form  of  tooth;  and  by  its  aid  exact  duplicates  may 
be  made  in  any  numbers  with  the  greatest  facility.  Its  general  appearance 
is  shown  in  Fig.  11.  It  will  be  noted  that  the  universal  joints  are  not 


Fig.  11.     Pantagraphic  Engine  for  forming  Cutters. 


49 

actually  of  the  ball  and  socket  kind,  which  suggests  the  explanation,  that 
in  Figs.  3-10  inclusive,  we  have  made  no  attempt  to  give  precise  details 
or  proportions,  but  only  to  make  as  clear  as  we  are  able  to,  the  principles 
and  mode  of  action  of  these  remarkably  ingenious  machines,  as  well  as  of 
the  system  adopted  in  using  them. 

THEORETICAL    DEFECTS    OF    THE    SYSTEM. 

It  forms  no  part  of  our  plan  to  represent  as  perfect  that  which  is  not 
so,  and  there  are  one  or  two  facts,  which  at  first  thought  might  seem 
serious  objections  to  the  adoption  of  the  epicycloidal  system.  These  are; 


1.  It  is  physically  impossible  to  mill  out  a  concave  cycloid,*by  any 
means  whatever,  because  at  the  pitch  line  its  radius  of  curvature  is  zero, 
and  a  milling  cutter  must  have  a  sensible  diameter. 


50 

2.  It  is  impossible  to  mill  out  even  a  convex  cycloid  or  epicycloid,  by 
the  means  and  in  the  manner  above  described. 

This  is  on  account  of  a  hitherto  unnoticed  peculiarity  of  the  curve  at 
a  constant  normal  distance  from  the  cycloid.  In  order  to  show  this  clearly, 
we  have,  in  Fig.  12,  enormously  exaggerated  the  radius  CD,  of  the  milling 
cutter  (M  of  Figs.  7  and  8).  The  outer  curve  HL,  evidently,  could  be 
milled  out  by  the  cutter,  whose  center  travels  in  the  cycloid  CA;  it  re- 
sembles the  cycloid  somewhat  in  form,  and  presents  no  remarkable  Teatures. 
But  the  inner  one  is  quite  different ;  it  starts  at  D,  and  at  first  goes  down, 
inside  the  circle  whose  radius  is  CD,  forms  a  cusp  at  E,  then  begins  to  rise, 
crossing  this  circle  at  G,  and  the  base  line  at  f.  It  will  be  seen,  then,  that 
if  the  center  of  the  cutter  travel  in  the  cycloid  A  C,  its  edge  will  cut  away 
the  part  GED,  leaving  the  template  of  the  form  OGI.  Now  if  a  roller  of 
the  same  radius  CD,  be  rolled  along  this  edge,  its  center  will  travel  in  the 
cycloid  from  A,  to  the  point  P,  where  a  normal  from  G,  cuts  it ;  then  the 
roller  will  turn  upon  G  as  a  fulcrum,  and  its  center  will  travel  from  P  to 
C,  in  a  circular  arc  whose  radius  to  GP=  CD. 

That  is  to  say,  even  a  roller  of  the  same  size  as  the  original  milling 
cutter,  will  not  retrace  completely  the  cycloidal  path  in  which  the  cutter 
traveled. 

Now  in  making  a  rack  template,  the  cutter,  after  reaching  C,  travels 
in  the  reversed  cycloid  CR,  its  left-hand  edge,  therefore,  milling  out  a 
curve  DK,  similar  to  HL.  This  curve  lies  wholly  outside  the  circle  DI, 
and  therefore  cuts  OG  at  a  point  between  ^and  G,  but  very  near  to  G. 
This  point  of  intersection  is  marked  S  in  Fig.  13,  where  the  actual  form  of 
the  template  OSK  is  shown.  The  roller  which  is  run  along  this  template, 
is  larger,  as  has  been  explained,  than  the  milling  cutter.  When  the  point 
of  contact  reaches  S  (which  is  so  near  to  G  that  they  practically  coincide), 
this  roller  cannot  now  swing  about  S  through  an  angle  so  great  as  PGC  of 
Fig.  12 ;  because  at  the  root  D,  the  radius  of  curvature  of  DK  is  only 


51 

equal  to  that  of  the  cutter,  and  G  and  6"  are  so  near  the  root  that  the  curva- 
ture of  SK,  near  the  latter  point,  is  greater  than  that  of  the  roller.  Con- 
sequently there  must  be  some  point  U  in  the  path  of  the  center  of  the 
roller,  such,  that  when  the  center  reaches  it,  the  circumference  will  pass 
through  S,  and  be  also  tangent  to  SK.  Let  T  be  the  point  of  tangency ; 
draw  SC7 and  TU,  cutting  the  cycloidal  path  AR  in  JSf  and  K  Then,  UY 
being  the  radius  of  the  new  milling  cutter  (corresponding  to  N  oi  Fig.  9), 
it  is  clear  that  in  the  outline  of  the  gear  cutter  shaped  by  it,  the  circular 
arc  .ATKwill  be  substituted  for  the  true  cycloid. 

THE    SYSTEM    PRACTICALLY    PERFECT. 

The  above  defects  undeniably  exist ;  now,  what  do  they  amount  to  ? 
The  diagrams,  Figs.  12  and  13,  are  drawn  purposely  with  these  sources  of 
error  greatly  exaggerated,  in  order  to  make  their  nature  apparent  and  their 


FIG.  14.      SET  OF  WHEELS  AND  RACK. 

existence  sensible.  The  diameters  used  in  practice,  as  previously  stated, 
are:  describing  circle,  7£  inches;  cutter  for  shaping  template,  \  of  an  inch; 
roller  used  against  edge  of  template,  H  inches;  cutter  for  shaping  a  I -pitch 
gear  cutter,  1  inch. 

l7NivERSt*r?  OP  CALIFORNIA 

SANTA  BARBARA  COLLEGE  L 


52 

With  these  data  the  writer  has  found  that  the  total  length  of  the  arc 
XY  of  Fig.  13,  which  appears  instead  of  the  cycloid  in  the  outline  of  a 
cutter  for  a  1-pitch  rack,  is  less  than  0.0175  inch;  the  real  deviation  from 
the  true  form,  obviously,  must  be  much  less  than  that.  It  need  hardly  be 
stated  that  the  effect  upon  the  velocity  ratio  of  an  error  so  minute,  and  in 
that  part  of  the  contour,  is  so  extremely  small  as  to  defy  detection.  And 
the  best  proof  of  the  practical  perfection  of  this  system  of  making  epicy- 
cloidal  teeth  is  found  in  the  smoothness  and  precision  with  which  the-wheels 
run ;  a  set  of  them  is  shown  in  gear  in  Fig.  14,  the  rack  gearing  as 
accurately  with  the  largest  as  with  the  smallest.  To  which  is  to  be  added, 
finally,  that  objection  taken,  on  whatever  grounds,  to  the  epicycloidal  form 
of  tooth,  has  no  bearing  upon  the  method  above  described  of  producing 
duplicate  cutters  for  teeth  of  any  form,  which  the  pantagraphic  engine  will 
make  with  the  same  facility  and  exactness,  if  furnished  with  the  proper 
templates. 


Table  of  Cutters  for  Teeth  of  Gear  Wheels, 

— MADE   BY — 

THE   PRATT  &  WHITNEY  CO.,  HARTFORD,  CONN.,  U.  S.  A 


All  Gears  of  the  same  pitch  cut  with  our  Cutters  are  perfectly  interchangeable. 


Diameter  of 
Cutters. 

Diametral 
Pitch. 

Price  of  Cutters. 

Size  of  Hole 
in  Cutters. 

SET  OF  24  CUTTERS. 

For  each  pitch  coarser  than  10. 

inches. 

11 

$25  00 

1}  inches. 

No.    1  cuts                   12  T 

9 

20  00 

a         u 

No.    2    "                      13 

" 

2J 

18  00 

"         " 

No.    3    "                      14 

3 

15  00 

"         " 

No.    4    "                      15 

3£ 

12  00 

1 

No.    5    "                       16 

4 

9  00 

«         u 

No.    6  '  "                      17 

5 

7  00 

it         « 

No.    7    "                      18 

•« 

6 

6  00 

((                   U 

No.    8     "                       19 

" 

7 

5  00 

(I                  U 

No.    9    "                       20 

« 

8 

4  50 

I     ;; 

No.  10    "                  21  to  22 

" 

9 

4  00 

No.  11     "                  23  "  24 

" 

10 

3  50 

"     « 

No.  12    "                 25  "  26 

" 

12 

3  50 

«(     « 

No.  13    "           '      27  "  29 

r        " 

14 

3  50 

«     « 

No.  14    "                  30  "  33 

r     ;;   . 

16 

3  00 

«     ti 

No.  15     "                  34  "  37 

18 

3  00 

u     « 

No.  16    "                  38  "  42 

i     « 

20 

3  00 

"     " 

No.  17    "                  43  "  49 

1   ;; 

22 

3  00 

«              u 

No.  18    "                 50  "  59 

24 

3  00 

"         " 

No.  19    "                 60  "  75 

" 

26 

3  00 

«         « 

No.  20    "                 76  "  99 

" 

28 

3  00 

"         " 

No.  21     "                100  "  149 

tt 

30 

3  00 

"         " 

No.  22    "                150  "  299 

« 

32 

3  00 

"         " 

No.  23    "               300  Rack. 

No.  24    "                    Rack. 

The  Cutters  are  made  for  diametrical  pitches.  By  diametrical  pitch  is  meant,  the 
number  of  teeth  per  inch  in  the  diameter  of  the  gear  at  pitch  line.  Two  pitches  should 
always  be  added  to  this  diameter  in  preparing  a  gear  for  cutting.  For  example :  a  gear  of 
80  teeth,  8  to  the  inch,  diametrical  pitch,  would  be  10  inches  on  pitch  circle,  but  the  gear 
should  be  turned  10  2-8  (or  J).  The  teeth  should  always  be  cut  two  pitches  deep  beside 
clearance. 

The  Cutters  are  made  for  a  clearance  of  1-16  of  the  depth  of  the  tooth :  example :  8 
to  the  inch  has  a  clearance  of  1-64;  therefore  the  tooth  should  be  cut  two  pitches  (1-4) 
and  1-64  deep.  The  gears  must  be  set  to  run  with  this  clearance  to  give  the  best  results. 

In  ordering  bevel  gear  cutters,  give  the  diameter  of  gear  at  outside  pitch  line,  and 
number  of  teeth,  also  the  width  of  face.  For  the  present  all  cutters  are  made  to  order. 


THE   PRATT  -&  WHITNEY  COMPANY, 

HARTFORD,  CONN., 

MANUFACTURERS    OF 

js/n^oHiisrisTS'  TOOLS 

for  general  use,  comprising  a  large  variety  of  Lathes,  Planers,  Drilling,  Milling,  Boring, 

Screw  making,  Bolt  cutting,  Die  sinking,  Grinding,  Polishing,  Shaping,  Tapping 

and    Marking   Machines,    Planer  and   Milling  Machine   Vises,    Planer, 

Milling  Machine  and  Bench  Centers,  Cam  cutting  Machines  for 

various  purposes,  Power  Shears,  a  variety  of  Power 

and  Foot  Presses,  Iron  Cranes  for  shops  and 

other  purposes,  Lathe  Chucks,  etc. 

MACHINES  FOR  GUN  AND  SEWING  MACHINE  MAKERS  AND  FOR 
ALL  USES  IN  METAL  WORKING. 

Having  furnished  several  plants  complete  for  the  manufacture  of  Guns,  Pistols,  Sewing 

Machines,  etc.,  we  particularly  solicit  such  business,  and  where  only  drawings  or 

models  are  furnished,  are  prepared  to  complete  such  tools  and  machines 

as  may  be  required,  and  to  send  competent  men  to  superintend 

their  erection,  and  to  run  them  if  required. 


A^ND     DIES 

Special   attention  given  to  this  branch,  and   in  form   of  thread,  mathematical  exactness 

and  workmanship,  are  unsurpassed,  and  have  become  the  "  standard  " 

in  many  leading  workshops. 

Forging  Machinery, 

Consisting  of  DROP  HAMMERS  (a  specialty)  in  six  sizes,  of  best  and  most  modern  con- 

struction; TRIP  HAMMERS,  TRIMMING  PRESSES,  SHEARS,  etc., 

FORGES  and  DROP  HAMMER  DIES. 

CUTTERS  FOR  GEAR  WHEELS, 

made  by  new  process  and  entirely  by  machinery,  which  secures  absolute  correctness  as  to 
form  and  interchangeability  of  any  given  pitch. 

SPECIAL  MACHINERY  (to  order). 

Are  prepared  to  perfect  plans  and  models  for  same.     Our  facilities  for  the  above,  and  for 

the  different  branches  of  Forging,  Finishing,  Casting,  etc.,  are  unsurpassed. 
IVIany    of    the     above     Tools    and    Machines    are    kept   in    stock. 

Illustrated  Catalogue  and  Price  Lists  furnished  on  application. 

THE  PRATT  &  WHITNEY  CO.,  Hartford,  Conn. 


UNIVERSITY  OF  CALIFORNIA 

Santa  Barbara  College  Library 
Santa  Barbara,  California 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


LD  21-10m-10,'ol 
(8066s4)476 


000  590  362 


TJ 

136 

M3 


